KAM Theory And The Application In The Existence Of Quasi-periodic Solutions For PDEs

In Chapter 1, we introduce the historical background, some recent results of KAM theory obtained in the literature and our main work in this paper.In Chapter 2, we consider the time dependent Schrodinger operator with a temporal quasi-periodic unbounded perturbation where P(φ) is an operator valued function which is Ce on the n-dimensional tori Tn= Rn/2πZn. For any l≥100(3n+2Υ+1), we construct a reduction theory of KAM type for the above operator, by which we prove that the quasi-periodic solutions for the Hamiltonian partial differential equations with unbounded and"limit" perturbations are linearly stable.In Chapter 3, we consider a perturbed KdV equation with higher order nonlinearity subject to periodic conditions. By searching a partial Birkhoff normal form up to order 6 and applying KAM theorem in Kappeler-Poschel [34] with some modifications, we obtain that there exist large quantities of 2-dimensional invariant tori, and thus quasi-periodic solutions for the above equation.